In mathematics, more specifically in functional analysis, a Banach space is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit that is within the space.
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members. The number of elements is called the length of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an index set that may not be numbers to another set of elements.
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue, although according to the Bourbaki group they were first introduced by Frigyes Riesz. Lp spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, finance, engineering, and other disciplines.
In mathematics, an infinite series of numbers is said to converge absolutely if the sum of the absolute values of the summands is finite. More precisely, a real or complex series is said to converge absolutely if for some real number . Similarly, an improper integral of a function, , is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if
In mathematics, a trace-class operator is a compact operator for which a trace may be defined, such that the trace is finite and independent of the choice of basis. Trace-class operators are essentially the same as nuclear operators, though many authors reserve the term "trace-class operator" for the special case of nuclear operators on Hilbert spaces and reserve "nuclear operator" for usage in more general topological vector spaces.
In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) such that the canonical evaluation map from into its bidual is an isomorphism of TVSs. Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space is reflexive if and only if the canonical evaluation map from into its bidual is surjective; in this case the normed space is necessarily also a Banach space. In 1951, R. C. James discovered a Banach space, now known as James' space, that is not reflexive but is nevertheless isometrically isomorphic to its bidual.
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces. All Banach and Hilbert spaces are Fréchet spaces. Spaces of infinitely differentiable functions are typical examples of Fréchet spaces, many of which are typically not Banach spaces.
In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field. In its basic form, it asserts that for a family of continuous linear operators whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm.
In functional analysis, a bounded linear operator is a linear transformation between topological vector spaces (TVSs) and that maps bounded subsets of to bounded subsets of If and are normed vector spaces, then is bounded if and only if there exists some such that for all in
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.
In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space.
In the mathematical field of analysis, the Nash–Moser theorem, discovered by mathematician John Forbes Nash and named for him and Jürgen Moser, is a generalization of the inverse function theorem on Banach spaces to settings when the required solution mapping for the linearized problem is not bounded.
In mathematical analysis, a Banach limit is a continuous linear functional :\ell ^{\infty }\to \mathbb {C} } defined on the Banach space of all bounded complex-valued sequences such that for all sequences , in , and complex numbers :
- (linearity);
- if for all , then (positivity);
- , where is the shift operator defined by (shift-invariance);
- if is a convergent sequence, then .
In mathematics, a nuclear space is a topological vector space that can be viewed as a generalization of finite dimensional Euclidean spaces that is different from Hilbert spaces. Nuclear spaces have many of the desirable properties of finite-dimensional vector spaces. The topology on them can be defined by a family of seminorms whose unit balls decrease rapidly in size. Vector spaces whose elements are "smooth" in some sense tend to be nuclear spaces; a typical example of a nuclear space is the set of smooth functions on a compact manifold.
In mathematics, there are usually many different ways to construct a topological tensor product of two topological vector spaces. For Hilbert spaces or nuclear spaces there is a simple well-behaved theory of tensor products, but for general Banach spaces or locally convex topological vector spaces the theory is notoriously subtle.
In mathematics, Mazur's lemma is a result in the theory of Banach spaces. It shows that any weakly convergent sequence in a Banach space has a sequence of convex combinations of its members that converges strongly to the same limit, and is used in the proof of Tonelli's theorem.
In the mathematical field of functional analysis, the space denoted by c is the vector space of all convergent sequences (xn) of real numbers or complex numbers. When equipped with the uniform norm:
In mathematics, , the vector space of bounded sequences with the supremum norm, and , the vector space of essentially bounded measurable functions with the essential supremum norm, are two closely related Banach spaces. In fact the former is a special case of the latter. As a Banach space they are the continuous dual of the Banach spaces of absolutely summable sequences, and of absolutely integrable measurable functions. Pointwise multiplication gives them the structure of a Banach algebra, and in fact they are the standard examples of abelian Von Neumann algebras.
In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector spaces (TVSs).
In functional analysis, every C*-algebra is isomorphic to a subalgebra of the C*-algebra of bounded linear operators on some Hilbert space H. This article describes the spectral theory of closed normal subalgebras of
